Hello! I've been intrigued by the following question for a while now and I was hoping you could help me out

Two cars are driving 36 km/h and are coming from opposite directions.
(like this: car 10m/s --> <-- car 10 m/s)
They each weigh 1000 kg. Then they crash. What is the relative speed and energy of this crash?

My thoughts:
I think most people will say the relative speed of the crash is 20 m/s, and therefore the energy in the crash is according to E=0.5*m*v^2,
= 0,5*1000*20^2 = 200000

But if you calculate the energy that each car individually has, it's only E = 0,5*1000*10^2, = 50000. So with two cars the total energy can only be 100000?

Yes, you're correct. It makes sense to consider the individual contributions of kinetic energy made by both cars using an external observer as the main reference frame.

Note that there's nothing wrong with calculating the kinetic energy using the relative speed and using one of the cars as a reference frame, but then all of the resulting dynamics (e.g. debris, deformation, etc.) would have to be calculated in the same moving reference frame using that energy budget, which might look a bit odd since the collision is likely going to decelerate both cars rapidly to near zero velocity after collision. For a head-on collision it's probably better to use the ground as the frame of reference.

They both travel at 10m/sec , if everything comes to rest your calculation is correct and the total energy expended is 100,000J

But you have raised an interesting point ... if your frame of reference is located on one of the cars , this car can be seen as stationary , the other car impacts at 20m/sec ...

So why isn't the energy released 200,000?? ...

Because your frame of reference is not allowed to change velocity , so the cars will crash and come to rest , but not rest with reference to your frame of reference , from that perspective both crashed stationary cars will be moving -10m/sec...

From this frame of reference the two cars still have kinetic energy but negative 100,000

So from the moving frame of reference energy is +200,000 and - 100,000 which makes 100,000 the same as from a stationary perspective..

Because your frame of reference is not allowed to change velocity , so the cars will crash and come to rest , but not rest with reference to your frame of reference , from that perspective both crashed stationary cars will be moving -10m/sec...

From this frame of reference the two cars still have kinetic energy but negative 100,000

You're quite right Oz, but I think you mean + 100,000 J, not -100,000 J.

Before: 200,000 J
After: 100,000 J

so 100,000 J has been exchanged between the two cars in the moving reference frame, just like the case for the stationary reference frame.

Negative energy is impossible. If the cars have -100,000 J after the crash, the total difference is 300,000 Joules. Furthermore, the equation for kinetic energy is

$\displaystyle KE = \frac{1}{2} m v^2$

So the only way to get negative kinetic energy is to have negative mass, which is not the case for cars.

I think if you read my post carefully it's OK ....

If the frame of reference is on one car we could be forgiven for thinking the energy at impact is 200,000 , but it can't be , and this is explained because the frame of reference is moving which gives an apparent KE of the crashed stationary cars of 100,000 in the other direction ...we have to subtract it from the 200,000

Hopefully other members will give their thoughts.

Let's look at things in another way .... our frame of reference is on one car , the other impacts at 20m/sec , but they will not come to rest in our frame of reference the whole crashed mass is travelling at 10m/sec

energy in colliding car at 20m/sec = 200,000

energy in moving mass of crashed two cars = 100,000

these two have to be subtracted to get the correct figure of 100,000

It's a mathematical convenience and is valid to have one figure negative .

It's a mathematical convenience and is valid to have one figure negative .

It's neither mathematically convenient nor valid to have a negative kinetic energy, it's incorrect. If you were talking about potential energy (e.g. electron energy levels in atoms) then you'd have a point, but not here.

Sorry to be pedantic about this, but I want to make sure the OP understands that you can't just go making energy values negative (at least without consequences), especially in macroscopic motion problems. They can lead to errors. For example, a confused student might try to find the velocity of an 1000 kg object with kinetic energy equal to -100,000 J...